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Subject: More skewb thoughts
From: mark.longridge@canrem.com (Mark Longridge)
Message-Id: <60.1030.5834.0C1CD83B@canrem.com>
Date: Fri, 3 Feb 1995 00:40:00 -0500
Organization: CRS Online  (Toronto, Ontario)

The following is a follow up to the discussion on the SKEWB
containing quotes from messages of Martin and myself.

>>  The number of positions both David Singmaster and Tony Durham
>>  (the inventor) find for the skewb is 3,149,280.

> Right.  The SKEWB has 75582720 basic states.  Just as with the cube,
> we consider two basic states to be essential equal if the differ only
> by a rotation of the rigid cube.  Since there are 24 rotations of
> the rigid cube, the SKEWB has 3149280 = 75582720/24 essential states.

I just noticed that the number of states of the pyraminx (with vertex
rotations included) also equals 75,582,720. (933,120 * 3^4)

>>Mark continued
>>
>>If we use only one tetrad of the skewb then GAP also finds this
>>  number:
>>
>>  ##    corners                                  centers
>>  ##    (each turn permutes 4)           (each turn permutes 3)
>>  skewb := Group(
>>      ( 1,11,17) ( 2,12,20)( 4,10,18)(22, 6,14) (25,27,29),
>>      ( 2,10,22) ( 1, 9,23)( 3,11,21)(17, 5,15) (25,27,30),
>>      ( 4,14,20) ( 1,15,19)( 3,13,17)( 7,11,23) (25,28,29),
>>      ( 6,12,18) ( 5,11,19)( 7, 9,17)(21, 1,13) (26,27,29)
>>  );;

I'll amend 'each turn permutes 4' to 'rotates one, 3-cycles the
others', although half the corners do move in some way. Also
the operators are RUF, RUB, RDF and lastly LUF.

The corner LDB remains fixed, so just like the 2x2x2 cube we are
fixing a corner.

>Note however, that the corners corresponding to the four generators for
>this subgroups do *not* form a tetrad.  They are the corner RUF and the
>three adjacent corners.

My computer Webster says that a tetrad is 'A group of four'. Perhaps
there is another meaning in geometry or group theory? Certainly I
agree with the 2nd statement.

>Snip< I concur with the Martin's next paragraph (excuse the editing)

>So allow me to use the subgroup H generated by RUF, LUB, RDB, and LDF.
>The corresponding four corners do form a tetrad.

Martin, could you clarify the use of tetrad here?  :-)

>More Snips<

>> Mr. Singmaster had indicated in his last Cubic Circular that we may
>> determine the skewb's orientation if only one of the tetrads are
>> moved.

> I am not certain that I understand what this means.   >Snip<

I'm going to re-read the article and think about this some more.

>> By moving first one tetrad and then the other we can easily change
>> the skewb's orientation in space.

> This comment I don't understand at all.  Could you clarify it a bit?

I shall amend by comment >> above to:

 By moving first one half of the puzzle and then the other we can
easily change the skewb's orientation in space.

 As stated in Douglas Hofstadter's column in the July 1982 issue of
Scientific American, the skewb is a deep-cut puzzle, that is
the part of the puzzle that is being operated on is no
smaller than the stationary portion.

-> Mark <-